Koszul Bicomplexes and Generalized Koszul Complexes in Projective Dimension One

Details Koszul Bicomplexes and Generalized Koszul Complexes in Projective Dimension One Koszul Bicomplexes and Generalized Koszul Complexes in Projective Dimension One

2005-10-27

arxiv.org/abs/math/0510598v1

Communications in Algebra 34, 4131 - 4156 (2006)

Mathematics

Commutative Algebra

Scientific paper

10.1080/00927870600876284

We describe Koszul type complexes associated with a linear map from any module to a free module, and vice versa with a linear map from a free module to an arbitrary module, generalizing the classical Koszul complexes. Given a short complex of finite free modules, we assemble these complexes to what we call Koszul bicomplexes. They are used in order to investigate the homology of the Koszul complexes in projective dimension one. As in the case of the classical Koszul complexes this homology turns out to be grade sensitive. In a special setup we obtain necessary conditions for a map of free modules to be lengthened to a short complex of free modules.

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